gmat CAT 1

30 people in total attended an office party for a colleague's birthday. The birthday cake was sliced into exactly 32 pieces, all of which were eaten. Did everyone who attended eat at least one slice of cake? (1) One person ate exactly 2 slices of cake. (2) One person ate exactly 3 slices of cake. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient. EACH statement ALONE is sufficient. Statements (1) and (2) TOGETHER are NOT sufficient.

(1) INSUFFICIENT: Since one person ate exactly 2 slices of cake, there are 30 slices for the remaining 29 people. It is possible, though not certain, that each of the remaining people ate at least one slice of cake. (2) INSUFFICIENT: Similar to (1), there are 29 slices of cake remaining for the other 29 people. It is possible, but not certain, that everyone else had exactly one slice of cake. (1) AND (2) SUFFICIENT: Combined we know that 2 of the people ate a combined 5 slices of cake. Thus there are only 27 slices remaining for the other 28 people, which is not enough for everyone to have his or her own slice of cake. The correct answer is C.

If $ defines a certain operation, is p $ q less than 20? (1) x $ y = 2x2 - y for all values of x and y (2) p = 4, q = 10

(1) INSUFFICIENT: This gives the definition of the $ function, however, it gives us no information about p and q. (2) INSUFFICIENT: This statement gives us no information about the $ function. (1) AND (2) SUFFICIENT: We can use the definition of the $ function given in (1) along with the values of p and q from (2) to solve for the value of p $ q = 2(4)2 - 10 = 22. The correct answer is C.

For each of the past two years, major department stores have reported a nearly 50% increase in their revenue generated from the sale of men's clothing manufactured by Zachary, Inc., a result that is all the more surprising because the sales of most other brands of men's clothing have been depressed over the same period. Nevertheless, Z.A.C., the parent company of Zachary Inc, does not appear to have emerged unscathed from the overall trend of decreased sales in the industry: Z.A.C. has reported a slight decline in overall sales in each of the past two years. Which of the following, if true, most helps to explain the surprising result above? The sales of clothing at Zachary, Inc.'s boutique stores, which, unlike department stores, are owned and operated by the company itself, have held steady over the last two years. Two years ago, Zachary, Inc. began an ambitious new advertising campaign; in each of the last two years, the company's advertising department has overspent its planned budget by almost half. Z.A.C. is renowned for the quality of its fabrics, and sells large quantities of fabric to a variety of manufacturers of men's clothing. Zachary, Inc. formerly manufactured leather accessories and women's clothing in addition to men's clothing, but, for the past three years, the company has produced only men's clothing. In the last two years, Z.A.C., in addition to maintaining its prior business ventures, expanded into two new markets, neither of which has been particularly profitable thus far.

(1) Identify the Question The prompt asks us to "explain the surprising result" in the passage, so this is an Explain the Discrepancy question. (2) Deconstruct the Passage According to the passage, sales of Zachary Inc men's clothing (a subsidiary of Z.A.C.) at department stores are up significantly over the last two years. Nevertheless, despite that increase, parent company Z.A.C.'s overall sales numbers have shrunk over the same period, in keeping with the trend for the sale of other lines of men's clothing. (3) State the Goal The "surprising result", here, is the decrease in Z.A.C.'s overall sales, despite the jump in the sales of its daughter company (Zachary Inc) men's clothing at department stores. Since the department-store sales of Zachary Inc's men's clothing have definitely increased, the only possible explanation for this result would be a decrease in some other aspect of Z.A.C.'s sales. (4) Work from Wrong to Right (A) According to this statement, the boutique stores' sales remained constant. To help explain the unexpected drop in the Z.A.C.'s overall sales, though, the boutique stores' sales would need to have decreased over the two-year period. (B) The current passage is focused only on Z.A.C. and Zachary Inc's sales, not on the overall profitability of Zachary Inc, so advertising expenses are irrelevant. (C) CORRECT. If sales of fabric to other men's clothing manufacturers - which themselves experienced depressed sales - constitutes a significant portion of Z.A.C.'s overall sales, then a decrease in that component of the Z.A.C.'s sales could conceivably account for the unexpected result in the passage. (D) The passage is concerned only with the trends in the Zachary Inc and Z.A.C.'s sales over the past two years, so any earlier changes in Zachary Inc's profile are irrelevant. (E) Participation in two new business ventures that have not proved to be profitable as of yet does not explain the decrease in sales reported for Z.A.C. Even if the lack of profit signifies poor sales in those two new areas, this lack of an increase in those two ventures does not necessarily constitute a decrease for the company overall. Furthermore, the answer choice also stipulates that Z.A.C. maintained its prior business ventures.

A newly discovered painting on wooden panel by Michelangelo must have been completed between 1507 and 1509. It cannot have been painted earlier than 1507 because one of its central figures carries a coin that was not minted until that year. It cannot have been painted after 1509 because it contains a pigment that Michelangelo is known to have abandoned when a cheaper alternative became available in that year. The answer to which of the following questions would be most useful in evaluating the argument above? Did any stocks of the abandoned pigment exist after 1509? Did Michelangelo work on the painting over the course of several years? Was the coin depicted in the painting in circulation in 1507? Can the wooden panel on which the painting was executed be tested accurately for age? Did Michelangelo's painting style change significantly between 1507 and 1509?

(1) Identify the Question Type The question stem asks what would be most useful in evaluating the argument, so this is an Evaluate the Argument question. (2) Deconstruct the Argument The author claims that the painting in question must have been completed between 1507 and 1509. What support is there for this claim? The part about 1507 seems fairly reasonable. How could Michelangelo paint a coin that did not exist yet? However, it's possible that Michelangelo had advance notice of what the new coin would look like. Perhaps he was shown the design in advance. He might even have designed the coin himself! The second restriction makes sense, too. If Michelangelo abandoned the pigment in 1509, then it shouldn't show up on his paintings after that point. However, this argument is specifically about when the painting was completed. Perhaps Michelangelo started with the old pigment and then finished in 1510 or later with the cheaper pigment. (3) State the Goal In an Evaluate the Argument question, the goal is to choose a question or piece of information that would make it easier to determine if the conclusion is valid. In this case, information about either of the two limiting dates would be useful. Did Michelangelo have advance notice about the coin? Did he start in one year and finish later? (4) Work from Wrong to Right (A) An answer of "yes" to this one might seem to cause trouble for the argument. Maybe Michelangelo still had the chance to use the more expensive pigment after 1509. However, the premise states definitively that Michelangelo abandoned that pigment sometime in 1509, and you do not want to contradict the premise! This answer choice would be helpful if the premise had said that the pigment was no longer produced, but that's not the issue. The pigment may well have been around after 1509, but Michelangelo wasn't using it. (B) CORRECT. This addresses the 1509 side of the conclusion. If Michelangelo worked on the painting for several years, he might have started with the more expensive pigment and then finished in 1510 or later with a different pigment. However, if he did not work on the painting for several years, then he must have completed it in 1509 or earlier, since he stopped using the expensive pigment after that year. (C) This is an interesting question, but it does not help to evaluate the conclusion. An answer of "yes" wouldn't impact the argument at all, as it's already clear that Michelangelo knew of the coin—he painted it! An answer of "no" would make it less likely that Michelangelo had seen the coin even in 1507, but if anything, this would just narrow the range further (maybe the coin became well known in 1508 or 1509). (D) It would certainly be helpful to test the painting for age. However, notice that like all of the answer choices in this problem, (D) is a yes/no question. A yes/no answer by itself won't help you to evaluate the author's conclusion. "Yes" just means that the claim can be tested scientifically, and "no" means that it can't. In order to evaluate, you would need to know the results of such a test! (E) This question is out of scope. The argument dates the painting between 1507 and 1509. Knowing that Michelangelo's style changed in that same period wouldn't make it any easier to tell if the painting was completed before 1507 (in the old style) or after 1509 (in the new style).

A list contains only integers. Are there more positive than negative integers in the list? (1) The median of the numbers in the list is positive. (2) The average (arithmetic mean) of the numbers in the list is positive. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient. EACH statement ALONE is sufficient. Statements (1) and (2) TOGETHER are NOT sufficient.

(1) NOT SUFFICIENT: Test numbers and try to disprove the question. For instance, the list 1, 2, 3, 4, 5 has a positive median (3) and contains more positive than negative integers. On the other hand, consider a list with an even number of entries—the median is halfway between two entries in the list, and not necessarily an entry in the list itself. For instance, the list -3, -2, -1, 101, 102, 103 has a positive median (50), but does not contain more positive than negative integers. (2) NOT SUFFICIENT: Test numbers and try to disprove the question. For instance, the list 1, 2, 3, 4, 5 has a positive median (3) and contains more positive than negative integers. On the other hand, consider a list containing small negative numbers and large positive numbers. For instance, the same list from statement 1 above, -3, -2, -1, 101, 102, 103, has a positive average, but does not contain more positive than negative integers. (In fact, unlike statement 1, this statement can also be satisfied by a list containing fewer positive than negative numbers, e.g., -5, -4, -3, -2, -1, 100). (1) AND (2) NOT SUFFICIENT: As deduced above, the lists 1, 2, 3, 4, 5 does have more positive integers while the list -3, -2, -1, 101, 102, 103 does not have more positive integers. Both lists satisfy both statements, so even when used together, the statements are still insufficient. The correct answer is E.

fraction equiv of .625

5/8

odd * even

= even

odd * odd

= odd

In the past year, there has been a dramatic increase in the number of people killed by alligators in Florida. During this same time, there has been an increase in the development of new houses, golf courses, and shopping areas in former wilderness areas within the state. Therefore, the increase in fatal alligator attacks must have been caused by the increase in the number of humans living in the alligator's habitat. Which of the following, if true, most seriously calls into question the explanation above? Two years ago, a government initiative to reduce the alligator population size by destroying alligator eggs ended. An increase in fatal alligator attacks tends to make people more cautious around lakes, ponds, swamps and canals. The number of people killed by snake bites, spider bites and scorpion stings in Florida has held steady for many years. Many of the new state residents have moved to newly constructed areas near water that is suitable for habitation by alligators. The undeveloped areas of Florida have decreased in area by 5% in the past year.

A conclusion may be weakened when another explanation at least as compelling as the original is offered. The number of people killed by alligators may have increased in the past year for some reason other than the increase in the number of humans living in the alligator's habitat. (A) CORRECT. This statement properly identifies an alternative rationale (there are more alligators now) and undermines the given explanation. (B) More cautious behavior would only occur after the fatal alligator attacks occurred, so it could not be a factor that supports or weakens the explanation. (C) This point about differing types of harmful wildlife is irrelevant to the argument about alligator fatalities. (D) This point could support the explanation if it could be proven that the increase in alligator attack fatalities were among these new residents. In any case, it does not weaken the explanation. (E) This point supports the explanation.

The population of locusts in a certain swarm doubles every two hours. If 4 hours ago the swarm just doubled to 1,000 locusts, in approximately how many hours will the swarm population exceed 250,000 locusts? 6 8 10 12 14 Hide Explanation

A population problem on the GMAT is best solved with a population chart that illustrates the swarm population at each unit of time. An example of a population chart is shown below: Time Population 4 hours ago 1,000 2 hours ago 2,000 NOW 4,000 in 2 hours 8,000 in 4 hours 16,000 in 6 hours 32,000 in 8 hours 64,000 in 10 hours 128,000 in 12 hours 256,000 As can be seen from the chart, in 12 hours the swarm population will be equal to 256,000 locusts. Thus, we can infer that the number of locusts will exceed 250,000 in slightly less than 12 hours. Since we are asked for an approximate value, 12 hours provides a sufficiently close approximation and is therefore the correct answer. The correct answer is D.

Line l is defined by the equation y - 5x = 4 and line w is defined by the equation 10y + 2x + 20 = 0. If line k does not intersect line l, what is the degree measure of the angle formed by line k and line w? (Assume that all lines lie in one coordinate plane.) 0 30 60 90 It cannot be determined from the information given. Hide Explanation

First, let's rewrite both equations in the standard form of the equation of a line: Equation of line l: y = 5x + 4 Equation of line w: y = -(1/5)x - 2 Note that the slope of line w, -1/5, is the negative reciprocal of the slope of line l. Therefore, we can conclude that line w is perpendicular to line l. Next, since line k does not intersect line l, lines k and l must be parallel. Since line w is perpendicular to line l, it must also be perpendicular to line k. Therefore, lines k and w must form a right angle, and its degree measure is equal to 90 degrees. The correct answer is D.

If ABCD is a trapezoid, where AB is parallel to CD, what is the area of ABCD? (1) 2(AB) + CD = 12 (2) The height of the trapezoid is 5 inches Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient. EACH statement ALONE is sufficient. Statements (1) and (2) TOGETHER are NOT sufficient.

First, recall that the area of a trapezoid is equal to the product of the height times the average of the two bases. Also, note that AB and CD are the bases of the trapezoid, since AB is parallel to CD. Therefore, to answer the question, we need to know the average of AB and CD, i.e. (AB + CD)/2 and the height. Finally, note that to find the average of AB and CD, it will be sufficient to find the sum AB + CD. Therefore, we can rephrase the question: "What is the height of the trapezoid and the sum AB + CD?" (1) INSUFFICIENT: This statement tells us nothing about either the sum AB + CD or the height of the trapezoid. To see that this statement cannot be manipulated to find the sum of the two bases, note that 2(AB) + CD = AB + (AB + CD) = 12, implying that (AB + CD) = 12 - AB, which depends on the value of AB. (2) INSUFFICIENT: This statement tells us the height of the trapezoid but provides no information about the sum of the bases. (1) AND (2) INSUFFICIENT: Even when the two statements are taken together, we know the height but still cannot find the sum of the two bases. All we know is that (AB + CD) = 12 - AB, which will depend on the value of AB. The correct answer is E.

Given distinct positive integers 1, 11, 3, x, 2, and 9, which of the following could be the median? 3 5 7 8 9

Median is the number in the middle of a sequence of numbers written in increasing order, assuming that the number of terms is odd. However, as we are given in this problem, if the number of terms is even, then we must average the two middle terms. The first task is to place the numbers in increasing order, making note of the fact that we don't know where to place x: 1, 2, 3, 9, 11 x? We're told our numbers are all distinct (or different) positive integers, which includes the possibilities for x. Given this, we know that the smallest number x can be is 4, and it could also be any larger integer except for 9 or 11. Some possibilities: 1, 2, 3, x, 9, 11 here, x could be 4, 5, 6, 7, or 8 1, 2, 3, 9, x, 11 here, x could be 10 1, 2, 3, 9, 11, x here, x could be 12, 13, or 14 In the first scenario, (3+x)/2 is the median, and since x could be 4, 5, 6, 7, or 8, the median could be 3.5, 4, 4.5, 5 or 5.5. The question asks us what could be the median. We've just found a possibility that appears among the answers: 5. If you're not sure, continue to check the other scenarios. In the remaining scenarios, 6 is the median - (3+9)/2. 6 is not an answer choice, however, so 5 must be correct. The correct answer is B.

If n is a non-negative integer such that 12^n is a divisor of 3,176,793, what is the value of n^12 - 12^n ? - 11 - 1 0 1 11 Hide Explanation

Since n must be a non-negative integer, n must be either a positive integer or zero. Also, note that the base of the exponent 12^n is even and that raising 12 to the nth exponent is equivalent to multiplying 12 by itself n number of times. Since the product of even integers is always even, the value of 12^n will always be even as long as n is a positive integer. For example, if n = 1, then 12^1 = 12; if n = 2, then 12^2 = 144, etc. Since integer 3,176,793 is odd, it cannot be divisible by an even number. As a result, if n is a positive integer, then 12^n (an even number) will never be a divisor of 3,176,793. However, if n is equal to zero, then 12^n = 12^0 = 1. Since 1 is the only possible divisor of 3,176,793 that will result from raising 12 to a non-negative integer exponent (recall that all other outcomes will be even and thus will not be divisors of an odd integer), the value of n must be 0. 0^12 - 12^0 = 0 - 1 = -1 The correct answer is B.

Everyone who has graduated from TopNotch High School has an intelligence quotient (IQ) of over 120. Most students with an IQ of over 120 and all students with an IQ of over 150 who apply to one or more Ivy League universities are accepted by at least one of them. The statements above, if true, best support which of the following conclusions? Every graduate of TopNotch High School with an IQ of 150 has been accepted by at least one Ivy-League university. If a person is a high-school graduate and has an IQ of less than 100, he or she could not have been a student at TopNotch High School. If a person has an IQ of 130 and is attending an Ivy-League university, it is possible for him or her to have graduated from TopNotch High School. At least one graduate from TopNotch high school who has applied to at least one Ivy-League university has been accepted to one of them. If a high-school graduate has an IQ of 150 and is not attending an Ivy-League university, then he or she did not apply to one of them.

Since the passage contains information about both TopNotch High School graduates and those accepted by Ivy League universities, a valid conclusion must contain information that does not contradict either situation. In addition, a valid conclusion should be based directly on the information given in the passage. Be careful not to make unwarranted assumptions: for example, a person who attends a school does not necessary graduate from it, a person who graduates from high school does not necessarily apply to a university, and a person who is accepted by a university does not necessarily attend it. (A) The passage states that every student with an IQ of 150 who applies to the Ivy League will be accepted by at least one of the universities. However, it is possible that a graduate of TopNotch High with an IQ of 150 did not apply, and thereby was not accepted, to any of the schools. Hence, this conclusion is not valid. (B) The passage states that every graduate of TopNotch High has an IQ of over 120. The conclusion only states that the student is a high school graduate and that he has an IQ of less than 120. It does NOT state that he or she was a graduate of TopNotch High. It is possible, however, that after attending TopNotch High for a period of time, he or she graduated from another high school. If this is the case, the situation does not contradict the passage, but contradicts the conclusion (he or she was a student at TopNotch High). Hence, this conclusion is not valid. (C) CORRECT. This choice presents a possibility, as opposed to a certainty: is it possible that a person with a 130 IQ and attending an Ivy League university previously graduated from TopNotch High School? This does fit the stated facts in the argument: TopNotch graduates all have IQs over 120, and it is possible for TopNotch graduates to gain acceptance to Ivy League universities. (D) The conclusion states that most, but not necessarily all, of the graduates from TopNotch High with IQ of 120 who apply to the Ivy League are accepted by at least one of the schools. The conclusion, however, does not state positively that any of the TopNotch High graduates had an IQ of over 150. Hence, even if it is unlikely, it is possible that none of the TopNotch graduates had IQs of over 150, and, of the remaining graduates who applied to the Ivy League, none were accepted by an Ivy League university. This conclusion is thereby not valid. (E) The passage states that any student with an IQ of 150 who applies to one or more Ivy League universities will be accepted by at least one of them. It is possible, however, that some of those who had applied and been accepted to an Ivy League university chose not to attend. Hence, this conclusion is not valid.

If n = 10^10 and n^n = 10^d, what is the value of d? 10^3 10^10 10^11 10^20 10^100

Substitute the given value of n into the right-hand equation, to give 10^10^(10^10) = 10^d Careful with the next step! The two exponents on the left may be multiplied: 10^(10*10^10) = 10^d Drop the two bases and set the exponents equal to each other: 10×10^10 = d Because the two bases are the same, you can add the two exponents: 10^11 = d The correct answer is C.

A store sells a certain product at a fixed price per unit. At the product's current price, q units cost a total of exactly $300. If the price were lowered by $5 from its current value, then q + 2n units would cost exactly $300; if the price were raised by $5, then q - n units would cost exactly $300. What is the value of q? 10 15 20 25 30

The answer choices are fairly "clean" (integers, not ridiculously large), so we can try plugging them into the problem until we find the one that works. Typically, when plugging the answers into the problem, we start with answer B or D and work from there. (Note: it is possible to use an algebraic solution here, but the algebra is so cumbersome that we can't recommend it. In fact, we're not even going to show it!) One other piece of advice: the math is complicated here, even to explain. Try writing out the steps below yourself as you read the explanation. (B) q = 15. If 15 units cost $300, then the fixed price is $20 per unit. If the price were lowered by $5, then the new price would be $15 and $300 would buy 300/15 = 20 units, so q + 2n = 20, or 15 + 2n = 20. This equation gives n = 2.5. If the price were raised by $5, then the new price would be $25 and $300 would buy 300/25 = 12 units. Check this against the final piece of information (q - n = the number of units). If B is the correct answer, we should be able to use the same value (2.5) for n and get the answer 12. 15 - 2.5, however, is 12.5, not 12, so B is not the correct answer. B was not correct but it was very close to correct (12.5 vs. 12), so the correct answer is likely to be either A or C. There is no easy way to tell which to try first; just pick one and try it. (A) q = 10. If 10 units cost $300, then the fixed price is $30 per unit. If the price were lowered by $5, then the new price would be $25 and $300 would buy 300 / 25 = 12 units, so q + 2n = 12, or 10 + 2n = 12. This equation gives n = 1. If the price were raised by $5, then the new price would be $35 and $300 would buy 300/35 = a non-integer number. We can't have a non-integer number of units, so A can't be the correct answer. (C) q = 20. If 20 units cost $300, then the fixed price is $15 per unit. If the price were lowered by $5, then the new price would be $10 and $300 would buy 300/10 = 30 units, so q + 2n = 30, or 20 + 2n = 30. This equation gives n = 5. If the price were raised by $5, then the new price would be $20 and $300 would buy 300/20 = 15 units. Check this against the final piece of information (q - n = the number of units). If C is the correct answer, we should be able to use the same value (5) for n and get the answer 15. Since q - n = 20 - 5 = 15, we know that q = 20 was the correct starting point. The correct answer is C.

Pascal has 96 miles remaining to complete his cycling trip. If he reduced his current speed by 4 miles per hour, the remainder of the trip would take him 16 hours longer than it would if he increased his speed by 50%. What is his current speed? 6 8 10 12 16 Hide Explanation

The best way to solve this problem is to Work Backwards from the Answer Choices, testing each answer choice in turn as follows: (A) 6mph Speed Time (T = D/R) Difference 4 mph slower 6 - 4 = 2 96 ÷ 2 = 48 50% faster 6 × 1.5 = 9 96 ÷ 9 = 10 More than 16! (B) 8 mph Speed Time (T = D/R) Difference 4 mph slower 8 - 4 = 4 96 ÷ 4 = 24 50% faster 8 × 1.5 = 12 96 ÷ 12 = 8 16 Once you test B, you can stop. C yields too small a difference in speed. (C) 10 mph Speed Time (T = D/R) Difference 4 mph slower 10 - 4 = 6 96 ÷ 6 = 16 50% faster 10 × 1.5 = 15 96 ÷ 15 = 6.4 9.6 To get a larger difference, look for a smaller initial speed, because a fixed change (4 miles per hour slower) makes more of an impact when the numbers are lower. As Pascal's speed increases, 4 miles per hour becomes smaller and smaller in relation to his overall speed, and therefore makes less of a difference. There is also an algebraic way to answer this question, although it is far more time consuming. The problem presents two hypothetical situations and compares them in terms of time. Therefore, begin by writing expressions to show the time each hypothetical trip would take. Label Pascal's current speed s, and express the time it would take him at the reduced speed as follows: Time = Distance/Rate = Similarly, if he increases his speed by 50%, then his time 150% of the original speed. Multiply s by 1.5: Time = Distance/Rate = Because we know that at reduced speed he would take 16 hours longer than at increased speed, we can relate the two times as follows: Slow time = Fast time + 16 = + 16 = + 16 = 96s = (s − 4) (64 + 16s) 96s = 64s + 16s2 − 256 − 64s 0 = 16s2 − 96s − 256 0 = s2 − 6s − 16 0 = (s − 8) (s + 2) s = +8 or -2 Because Pascal's current speed cannot be negative, the only possible value is 8 miles per hour. The correct answer is B.

If a , b, and c are integers and ab^2/c is a positive even integer, which of the following must be true? I. ab is even II. ab > 0 III. c is even I only II only I and II I and III I, II, and III

The fact that the quotient ab^2/c is even tells us that the numerator ab2 is even. If ab2 were odd, the quotient would never be divisible by 2, regardless of what c is. To prove this try to divide an odd number by any integer to come up with an even number; you can't. If ab2 is even, either a is even or b is even. (I) TRUE: Since a or b is even, the product ab must be even (II) NOT NECESSARILY: For the quotient to be positive, a and c must have the same sign since b2 is definitely positive. We know nothing about the sign of b. The product of ab could be negative or positive. (III) NOT NECESSARILY: For the quotient to be even, ab2 must be even but c could be even or odd. An even number divided by an odd number could be even (ex: 18/3), as could an even number divided by an even number (ex: 16/4). The correct answer is A.

City Controller: 63% of our residents voted to approve the developer's request to build a national chain hotel on the site of the old consignment store. The hotel will increase our revenue base and, therefore, provide more money for schools and community services. Mayor: But our recent survey showed that the most important reason people want to live here is our small-town feel resulting from the local ownership of the vast majority of businesses. What is the best explanation for the apparent contradiction in opinions cited by the controller and the mayor? Most people believe having a small-town feel is more important than having quality schools. A locally-owned business might be able to generate as much revenue as a well-known hotel chain. The recent survey did not ask about preferences for a chain hotel versus a locally-owned bed and breakfast. An increase in the town's revenue base may not result in additional money for the schools. The recent survey cited by the mayor polled people who are considering moving to the town.

The majority of residents voted to allow a national chain to build a hotel in town. This appears to contradict the mayor's claim that a recent survey showed most people want to live in the town because of the locally-owned businesses. In order to answer the question, we need an additional piece of information which explains away, or resolves, this apparent contradiction. (A) If people rank a small-town feel higher in importance than quality schools, then this fact would still contradict the combination of the vote and the survey response. (B) If a locally-owned business could generate the same amount of revenue as the national hotel, then this fact would still contradict the combination of the vote and the survey response. (C) This does not address why the residents voted to approve the hotel while survey respondents said they preferred locally-owned businesses. (D) This appears to contradict the reason given for residents voting to approve the hotel but does not do so in a way that explains the discrepancy with the survey respondents. (E) CORRECT. If the survey respondents were not actually residents of the town, this explains why the residents of the town voted in a way that does not reflect the results of the survey.

Unlike Mars, the surface of Earth is primarily water, with landmass making up less than half of the total area. Mars, the surface of Earth is primarily water Mars, Earth's surface is primarily water the surface of Mars, that of Earth is primarily water Mars, water is primarily Earth's surface that of Mars, Earth has a surface that is primarily water Hide Explanation

The original sentence begins with the comparison "unlike Mars." What follows must therefore be a logical comparison to the planet Mars. However, the sentence compares "Mars" to "the surface of Earth." This is not a logical comparison. We can compare "Mars" to "Earth" or "the surface of Mars" to "the surface of Earth," but it is not logical to compare one planet to the surface of another planet. (A) This choice is incorrect as it repeats the original sentence. (B) This choice compares "Mars" to "Earth's surface," an illogical comparison. (C) CORRECT. This choice compares "the surface of Mars" to "that of Earth," a logical comparison. (D) This choice compares "Mars" to "water," an illogical comparison. (E) This choice compares "that of Mars" to "Earth." In this context, it is not clear what "that of Mars" refers to, since there is no other possessive construction in the sentence.

Noting that the price of oil and other fuel components, a major factor in the cost structure of an airline, have risen and will continue to rise, the company management was pessimistic about their outlook for the upcoming quarter. have risen and will continue to rise, the company management was pessimistic about their have risen and will continue to rise, the company management was pessimistic about the will continue to rise, the company management was pessimistic about the has risen and will continue to rise, the company management was pessimistic about their will continue to rise, the company management was pessimistic about their

The original sentence supplies the plural verb construction "have risen" for the singular subject "price." Further, the phrase "have risen and will continue to rise" is redundant. Finally, the original sentence uses the plural pronoun "their" to refer to the singular subject "management." (A) This choice is incorrect as it repeats the original sentence. (B) This answer corrects the pronoun issue, but suffers from the lack of agreement between the subject "the price" and the verb "have risen." This answer choice also retains the redundant and wordy construction "have risen and will continue to rise." (C) CORRECT. This answer replaces the redundant construction "have risen and will continue to rise" with the more concise "will continue to rise." This change is possible without any loss of content, since using "will continue to rise" already implies that the price of oil and fuel components has been increasing to date. Further, this modification resolves the subject-verb agreement issue in the original sentence. Finally, this answer choice replaces the plural pronoun "their" with the article "the," thus remedying the original lack of agreement between the noun "management" and pronoun "their." (D) While supplying the appropriate singular verb "has risen" for the singular subject "the price," this choice is wordy and retains the incorrect pronoun "their" from the original sentence. (E) While resolving the issues of redundancy and subject-verb agreement, this answer uses the plural pronoun "their" to refer to the singular noun "management."

Many studies have shown that users of anabolic steroids exhibit habitual aggression and commit violent crime at rates significantly higher than those seen in the general public; the studies have claimed the existence of a "steroid rage," or "'roid rage," caused by the anabolic steroids themselves. This claim is mistaken, though, since individuals who elect to use anabolic steroids tend to be innately more aggressive than the general public. Which of the following, if true, most strengthens the argument above? Anabolic steroids produce significant increases in the levels of the hormones that are principally associated with aggressive behavior. Users of anabolic steroids must regularly "cycle off" the steroids; during these off-cycle times, their levels of aggression tend to decrease significantly. Individuals who choose to use anabolic steroids tend to have unusually high ambition and sex drives. Among people with identical histories of aggression, users of anabolic steroids do not exhibit significantly greater aggression than do non-users. Among individuals convicted of violent offenses while on anabolic steroids, most are not convicted again, even if they continue to use anabolic steroids.

The passage discounts the idea that anabolic steroid use causes aggressive behavior, stating that the causation actually runs in reverse: viz., a pre-existing tendency toward aggression motivates individuals to use anabolic steroids. (A) This statement, if true, weakens the argument considerably, because it provides strong support for the contention that the steroids themselves cause aggressive behavior. (B) This statement actually weakens the argument, because, if steroid users' aggression were due to an innate predisposition, then that aggression would be expected not to decline when steroid use is discontinued. (C) Although this statement is similar to the hypothesis in the argument - that individuals who choose steroids have different innate personality characteristics than do others - it deals only with sex drive and ambition. It does not deal with aggression, which is the sole focus of the passage, and is therefore irrelevant to the argument. (D) Correct This observation controls the variable cited as important in the passage - the individuals' history of aggression (which serves as an indicator of their innate tendency toward aggression) - and finds that, when that history is identical, anabolic steroids have essentially no effect on aggression. This is strong evidence for the fact that the tendency toward aggression itself, and not the steroid use, causes the aggression seen in steroid users. (E) This statement is irrelevant to the passage. First, convictions for violent offenses are not necessarily a good proxy for aggression itself. Second, the statement does not distinguish between convicts who continue steroid use and those who do not, a distinction that is key for resolving the issue presented in the passage.

If BD = CD, what is the degree measure of x + y? (1) v = 74 degrees (2) w = 32 degrees Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient. EACH statement ALONE is sufficient. Statements (1) and (2) TOGETHER are NOT sufficient.

The question asks us to find the number of degrees in x + y. Since the sum of the angles in any triangle is 180 degrees, we know that x + y + z = 180, so if we can find z, we can find x + y. Furthermore, z + v = 180, so if we can determine v, we can determine z. In fact, since angle v is an exterior angle to triangle ABD, we know that x + y = v, so we are really looking for v. Finally, since BD = CD, we know that triangle BCD is isosceles and that angle v equals angle u. Therefore we can rephrase the question as "What is the degree measure of angle v or angle u?". (1) SUFFICIENT: Statement (1) says that angle v = 74 degrees. This answers the rephrased question, so statement (1) is sufficient. (2) SUFFICIENT: Statement (2) says that w equals 32 degrees, so we can deduce the following, knowing that u and v are equal. w + u + v = 180 32 + 2v = 180 2v = 148 v = 74 This answers the rephrased question, so statement (2) is sufficient. The correct answer is D.

If x is a positive integer, is x prime? (1) x has the same number of factors as y2, where y is a positive integer greater than 2. (2) x has the same number of factors as z, where z is a positive integer greater than 2. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient. EACH statement ALONE is sufficient. Statements (1) and (2) TOGETHER are NOT sufficient.

The question stem tells us that x is a positive integer. Then we are asked whether x is prime; it is helpful to remember that all prime numbers have exactly two factors. Since we cannot rephrase the question, we must go straight to the statements. (1) SUFFICIENT: If x has the same number of factors as y2, then x cannot be prime. A prime number is a number that has only itself and 1 as factors. But a square has at least 3 distinct factors. For example, if y is prime, y = 2, then y2 = 4, which has 1, 2, and 4 as factors. If the root (in this case y) is not prime, then the square will have more than 3 factors. For example, if y = 4, then y2 = 16, which has 1, 2, 4, 8, and 16 as factors. In either case, x will have at least 3 factors, establishing it as nonprime. (2) INSUFFICIENT: If z is prime, then x will have only two factors, making it prime. But if z is nonprime, it will have more than two factors, which means x will have more than two factors, making x nonprime. Since we do not know which case we have, we cannot tell whether x is prime. The correct answer is A.

The consultant explained that companies that establish successfully operations abroad protect with consistency their intellectual property, lobby government officials without tiring, and empower local managers with aggression. that establish successfully operations abroad protect with consistency their intellectual property, lobby government officials without tiring, and empower local managers with aggression which establish operations abroad successfully protect intellectual property consistently, lobby government officials without tiring, and empower local managers aggressively that establish successful operations abroad consistently protect their intellectual property, lobby tirelessly government officials, and empower aggressive local management that successfully establish operations abroad consistently protect their intellectual property, tirelessly lobby government officials, and aggressively empower local managers of which operations abroad are successfully established protect their intellectual property consistently, lobby tirelessly government officials, and aggressively empower local management

The three listed behaviors of companies that successfully establish operations abroad are logically parallel; therefore, they should be structurally parallel. In the original sentence, the first activity "protect with consistency their intellectual property" is not structured in parallel fashion to the second and third activities. Also, the phrase "empower local managers with aggression" suggests that the local managers are being given the quality of aggression, which is not contextually appropriate; "aggression" is better applied to the act of empowering the managers, not to the managers themselves. (A) This choice is incorrect as it repeats the original sentence. (B) This choice incorrectly uses the relative pronoun "which." "Which" should be used for noun modifiers, whereas here "which" is incorrectly used to introduce a set of clauses that are integral to the sentence. Also, the second activity "lobby government officials without tiring" is not structured in parallel fashion to the first and third activities. (C) This sentence fails to follow an appropriate parallel structure as it lists the three activities of companies that are successful abroad. Also, the phrase "empower aggressive local managers" suggests that the local managers are aggressive which is not contextually appropriate; "aggressive" is better applied to the act of empowering the managers, not to the managers themselves. (D) CORRECT. The three logically parallel activities in this sentence are structurally similar and the sentence is clear and concise. (E) This choice incorrectly begins with the phrase "of which", which suggests that the operations of the company may be established abroad by a third party other than the company itself. Also, the three logically parallel activities are not structured in parallel fashion.

At a restaurant, a group of friends ordered four main dishes and three side dishes at a total cost of $91. The prices of the seven items, in dollars, were all different integers, and every main dish cost more than every side dish. What was the price, in dollars, of the most expensive side dish? (1) The most expensive main dish cost $16. (2) The least expensive side dish cost $10. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient. EACH statement ALONE is sufficient. Statements (1) and (2) TOGETHER are NOT sufficient.

There are seven different dishes (4 main dishes and 3 side dishes), each with a different integer cost; we cannot use just two variables to represent the two types of dishes. Each of the four main dishes is more expensive than any of the side dishes. The seven dishes add up to $91. The question asks for the price of the most expensive side dish. This is a statistics question, so it will be useful to consider different cases. (1) SUFFICIENT: Since this problem involves 7 different integers, some of which must be larger than others, it's a good idea to test extreme possibilities. We're told that the most expensive main dish is $16 so try maximizing the cost of the other three: $15, $14, and $13. In this case, the total cost of the four main dishes is $58, leaving $33 to split among the three side dishes. The prices of the three side dishes must be different integers and all must cost less than $13. Again, try maximum values first: $12, $11 and $10 are the largest possible prices for the three side dishes and these indeed sum to $33. However, note that this maximized case is the only case that will work - otherwise the 3 side dishes will not sum to $33 (and the total will not sum to $91). If we back up and try something other than the maximum values for the four main dishes, the main dishes will then sum to less than $58 and the lowest price of these main dishes will be something less than $13. This will mean we'll need the sum of the three side dishes to be greater than $33, and we'll have to achieve that with numbers smaller than 12, 11, and 10. That's impossible! Thus, $12 is the price of the most expensive side dish. Conceptually, we could have started with the average of these 7 dishes that sum to $91, that is $91/7 or $13. With the greatest statistic at $16 (only $3 above the mean), we might have understood that the other 6 prices can't be that far away from the mean (the mean is the balancing point for all of the statistics in the set). The set of consecutive integers from $16 to $10 would have been the first to try. After that worked (13 is the mean of these 7 integers and 7 x 13 = 91), we could have proved that all other cases would decrease the sum to less than $91. (2) SUFFICIENT: Since this problem involves 7 different integers, some of which must be larger than others, it's a good idea to test extreme possibilities. We're told that the least expensive side dish costs $10, so this time try minimizing the cost of the other two: $11, and $12. The side dishes would then cost a total of $33, leaving $58 for the main dishes. Since the minimum values for each of the four side dishes, i.e. $13, $14, $15, and $16 already sum to $58 there is no way to choose any larger values for these prices, which also means that we also couldn't have chosen any larger values for the price of the side dishes. Thus, $12 is the price of the most expensive side dish. Conceptually, we could have started with the average of the 7 dishes that sum to $91, that is $91/7 or $13. With the smallest statistic at $10 (only $3 below the mean), we might have understood that the other 6 prices can't be that far away from the mean (the mean is the balancing point for all of the statistics in the set). The set of consecutive integers from $16 to $10 would have been the first to try. After that worked (13 is the mean of these 7 integers and 7 x 13 = 91), we could have proved that all other cases would increase the sum to greater than $91. Note that this is a C-trap. Knowing that the greatest and smallest values of the set of 7 different integers were only 6 apart would imply that they must be consecutive integers. However, it was possible to get the answer with each statement alone. The correct answer is (D).

4.896/(1/0.07 + 1/0.16) is approximately equal to: 0.238 0.262 0.625 0.649 6.25

This question is best solved by approximating each of the elements. Let's refer to 4.896 as " less than 5". The first fraction in the denominator can be rewritten as follows: 1 .07 = 1 7 100 = 100 7 This can approximated as " more than 14," since 14 × 7 = 98 (i.e., 7 goes into 100 slightly more than 14 times). The second fraction in the denominator can be rewritten as follows: 1 .16 = 1 16 100 = 100 16 This can be approximated as " more than 6" since 16 × 6 = 96 (i.e., 16 goes into 100 slightly more than 6 times). " More than 14" + " more than 6" gives us " more than 20" in the denominator. Thus we have: less than 5 more than 20 The value of the fraction 5/20 = 0.25. Since the above fraction diminishes the numerator slightly (which has the effect of decreasing the fraction) and increases the denominator slightly (which also has the effect of decreasing the fraction), the value of the fraction should be slightly smaller than 0.25. The only possible answer choice is A.

comparisons....

This sentence contains two comparisons. It compares crocodiles and alligators in terms of their ability to tolerate high salinity; it also compares the two animals' ability (or lack thereof) to expel salt through the glands on their tongues. These comparisons must be written logically, comparing elements that are actually parallel to each other. In addition, the sentence must not use "compared to/with" with another term of comparison, such as better. ("Compared to/with" is used in sentences that simply state the two things being compared, without using any other words of comparison, e.g., The unemployment rate in County X is 5%, compared to 8% in County Y.)

Bag A contains red, white and blue marbles such that the red to white marble ratio is 1:3 and the white to blue marble ratio is 2:3. Bag B contains red and white marbles in the ratio of 1:4. Together, the two bags contain 30 white marbles. How many red marbles could be in bag A? 1 3 4 6 8 Hide Explanation

We are told that bag B contains red and white marbles in the ration 1:4. This implies that WB, the number of white marbles in bag B, must be a multiple of 4. What can we say about WA, the number of white marbles in bag A? We are given two ratios involving the white marbles in bag A. The fact that the ratio of red to white marbles in bag A is 1:3 implies that WA must be a multiple of 3. The fact that the ratio of white to blue marbles in bag A is 2:3 implies that WA must be a multiple of 2. Since WA is both a multiple of 2 and a multiple of 3, it must be a multiple of 6. We are told that WA + WB = 30. We have already figured out that WA must be a multiple of 6 and that WB must be a multiple of 4. So all we need to do now is to test each candidate value of WA (i.e. 6, 12, 18, and 24) to see whether, when plugged into WA + WB = 30, it yields a value for WB that is a multiple of 4. It turns out that WA = 6 and WA = 18 are the only values that meet this criterion. Recall that the ratio of red to white marbles in bag A is 1:3. If there are 6 white marbles in bag A, there are 2 red marbles. If there are 18 white marbles in bag A, there are 6 red marbles. Thus, the number of red marbles in bag A is either 2 or 6. Only one answer choice matches either of these numbers. The correct answer is D.

8^a(1/4)^b = ? (1) b = 1.5a (2) a = 2 Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient. EACH statement ALONE is sufficient. Statements (1) and (2) TOGETHER are NOT sufficient.

We can simplify the question as follows: 8^a(1/4)^b = ? [Break all non-primes down to primes.] (2^3)^a(2^-2)^b = ? [Multiply exponents taken on the same base.] (2^3a)(2^-2b) = ? [Add exponents since the two bases are equal.] 2^(3a - 2b) = ? We can rephrase the question as "what is 3a -2b?" (1) SUFFICIENT: b = 1.5a, so 2b = 3a. This means that 3a - 2b = 0. (2) INSUFFICIENT: This statement gives us no information about b. The correct answer is A.

Machine A can complete a certain job in x hours. Machine B can complete the same job in y hours. If A and B work together at their respective rates to complete the job, which of the following represents the fraction of the job that B will not have to complete? x - y/ x + y x/ y - x x + y/ xy y/ x - y y/ x + y Hide Explanation

We can solve this problem as a VIC (Variable In Answer Choice) and plug in values for the two variables, x and y. Let's say x = 2 and y = 3. Machine A can complete one job in 2 hours. Thus, the rate of Machine A is 1/2. Machine B can complete one job in 3 hours. Thus, the rate of Machine B is 1/3. The combined rate for Machine A and Machine B working together is: 1/2 + 1/3 = 5/6. Using the equation (Rate)(Time) = Work, we can plug 5/6 in for the combined rate, plug 1 in for the total work (since they work together to complete 1 job), and calculate the total time as 6/5 hours. The question asks us what fraction of the job machine B will NOT have to complete because of A's help. In other words we need to know what portion of the job machine A alone completes in that 6/5 hours. A's rate is 1/2, and it spends 6/5 hours working. By plugging these into the RT=W formula, we calculate that, A completes (1/2)(6/5) = 3/5 of the job. Thus, machine B is saved from having to complete 3/5 of the job. If we plug our values of x = 2 and y = 3 into the answer choices, we see that only answer choice E yields the correct value of 3/5.

What is the sum of the digits of the positive integer n where n < 99? 1) n is divisible by the square of the prime number y. 2) y4 is a two-digit odd integer. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient. EACH statement ALONE is sufficient. Statements (1) and (2) TOGETHER are NOT sufficient.

We cannot rephrase the given question so we will proceed directly to the statements. (1) INSUFFICIENT: n could be divisible by any square of a prime number, e.g. 4 (22), 9 (32), 25 (52), etc. (2) INSUFFICIENT: This gives us no information about n. It is not established that y is an integer, so n could be many different values. (1) AND (2) SUFFICIENT: We know that y is a prime number. We also know that y4 is a two-digit odd number. The only prime number that yields a two-digit odd integer when raised to the fourth power is 3: 34 = 81. Thus y = 3. We also know that n is divisible by the square of y or 9. So n is divisible by 9 and is less than 99, so n could be 18, 27, 36, 45, 54, 63, 72, 81, or 90. We do not know which number n is but we do know that all of these two-digit numbers have digits that sum to 9. The correct answer is C.

Set A contains three different positive odd integers and two different positive even integers; set B contains two different positive odd integers and three different positive even integers. If one integer from set A and one integer from set B are chosen at random, what is the probability that the product of the chosen integers is even? 6/25 2/5 1/2 3/5 19/25

We have several options for solving this problem. The most efficient way is via the 1 - x Strategy: that is, calculating the probability of the outcome that we do not want (odd) and subtracting from 1. This strategy is most efficient on this problem because there is only one way in which the product of 2 numbers can be odd: when the two starting numbers are also odd. If one or both of the starting numbers are even, then the product will also be even. 1 - x Strategy There are 3 possibilities for choosing an odd number from from set A and 2 possibilities for choosing an odd number from set B. There are 3 × 2 = 6 possibilities, then, for obtaining an odd product. The total number of general possible outcomes is 5 × 5 = 25, so the probability of obtaining an odd result is 6/25. Don't forget the last step! 1 - 6/25 = 25/25 - 6/25 = 19/25. Regular probability Alternatively, we can also solve directly for the number of even cases, but the calculations will be longer because we will have to account for three cases: (1) Odd integer from set A, even integer from set B: 3 × 3 = 9 different possibilities (2) Even integer from set A, odd integer from set B: 2 × 2 = 4 different possibilities (3) Even integer from set A, even integer from set B: 2 × 3 = 6 different possibilities There are 9 + 4 + 6 = 19 different ways to obtain an even product, so the desired probability is 19/25. Make a list or guess As a third option, we can see from the answers that there are at most 25 possibilities, so we could also list out the different possibilities and count them up. This will likely take at least the full 2 minutes, so we have to decide whether we want to spend that time, particularly when it would be easy to make a mistake with such a list. Alternatively, we could also realize that there is a higher probability of an even outcome than of an odd outcome (because we need only one even number to make the product even), so the probability should be greater than 1/2. That allows us to guess between answers D and E. If we decide to list out possibilities, we want to select particular integers in order to make the task more concrete. For example, let's say that set A = {1, 2, 3, 4, 5} and set B = {6, 7, 8, 9, 10} (make sure that the numbers you choose fit the criteria given in the problem). Make a table of all possible products (note that it is unnecessary to calculate the actual products; just determine whether they're even): 1 × 6 1 × 7 1 × 8 1 × 9 1 × 10 2 × 6 2 × 7 2 × 8 2 × 9 2 × 10 3 × 6 3 × 7 3 × 8 3 × 9 3 × 10 4 × 6 4 × 7 4 × 8 4 × 9 4 × 10 5 × 6 5 × 7 5 × 8 5 × 9 5 × 10 Of these 25 possible products, nineteen (shown in boldface) are even, so the desired probability is 19/25. The correct answer is E.

If a is divisible by 2, is b + 5 an integer? (1) The median of a and b is not an integer. (2) The average (arithmetic mean) of 3a, b, and b + 10 is an even integer. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient. Both statements TOGETHER are sufficient, but NEITHER one ALONE is sufficient. EACH statement ALONE is sufficient. Statements (1) and (2) TOGETHER are NOT sufficient.

With all Data Sufficiency problems, we start by rephrasing the question. A number that is divisible by 2 must be an even integer, and b + 5 is an integer whenever b is an integer. Our rephrased question is therefore: if a is even, is b an integer? (1) INSUFFICIENT: The median of two numbers is the average of those two numbers. Knowing that the average of a and b is not an integer does not help us determine whether b is an integer. A good way to test this statement is to pick numbers and prove insufficiency by showing that the median for a and b can be a non-integer using both integer ("Yes" to the question) and non-integer ("No" to the question) values of b. First we choose an integer value for b: let a = 2 and b = 3. The average of 2 and 3 is 2.5, a non-integer. This is a "Yes" to the original question: in this case, b is an integer. Next we deliberately try to choose a non-integer value of b that will also cause the median of a and b to be a non-integer. Let a = 2 and b = 3.5. The average is 2.75, a non-integer. This answers "No" to the original question: in this case, b is not an integer. (2) SUFFICIENT: Picking numbers for this statement proves to be a much more arduous task due to the heightened constraints: three variable terms each with coefficients or added constants and the restriction that the average be even. Therefore it is more advisable to use the Average Formula and look for a pattern. Plugging in we find that , where n represents an even integer. (3a + b + b + 10)/3= n Manipulating this equation, we get b = 3n/2 - 3a/2 - 10/2 From this equation we can assess that b must be an integer because every piece on the right-hand-side of the equation is an integer. 3n/2 must be an integer because n is an even integer. 3a/2 must be an integer because a is an even integer. And is equal to the integer 5. Therefore, b must be an integer, because adding or subtracting integers results in an integer. The correct answer is B.

Venn Diagram formula for when there are two individual options, a both option and/or a neither option

X + Y + Neither - Both = Total (ex. Spanish + French + Neither - Both = Total)

quadratics?

always look for them!

Area of a trapezoid

average of 2 bases * height of trap

how to tackle reading comp

breakdown main point of each paragraph

to find median on a chart

count number of independent variables, middle number will be the median one and go from largest to smallest by bar to find what that is

how to find standard deviation on a chart

distance of the set from its mean value, or the "spread" of the set. you can judge by eye. bigger if it moves around a ton from its mean

when answers are fairly clean integers...

first start by plugging in B and D and work from there

less vs fewer

less can only be used for noncountable nouns like "less water" or "less evidence". countable nouns such as "engineers" must use "fewer

semicolon proper use

the parts of the sentence before and after it must both be independent clauses (each can stand alone as a sentence).

when we have overlapping set question (ex. 2 color options and 2 size options)...

use a double set matrix to organize!